Abstract
Symmetry groups generated by charges which obey Gauss’ law are discussed as characteristic features of local gauge quantum field theories. The existence of such symmetry groups in a local quantum field theory leads to very important properties. In particular, unbroken charges obeying Gauss’ law always define superselection rules and their breaking always leads to the Higgs phenomenon, which has therefore a quite general explanation, related to Gauss’ law, without the introduction of Higgs fields.
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Notes and references
See e.g. Weinberg, S., Rev. Mod. Phys. 46, 255 (1974).
Yang, C. N., and Mills, R. L., Phys. Rev. 96, 191 (1954).
This is not only a pedagogical question, but also a crucial one if one wants to investigate general properties of GQFT without relying on perturbation theory, Lagrangian functions etc.; one of the main strategies of general QFT is in fact that of extracting basic properties of Lagrangian QFT, so that one may discuss their relations and their consequences in general.
For a more precise meaning of Equation (1) see following sections and Ref. 5.
Strocchi, F., Phys. Rev. D17, 2010 (1978) and references therein to previous papers.
It need not be stressed here the importance of such a local conservation law (for the charge density), which still retains a meaning even in the case of spontaneously broken symmetry, when the global conservation law fails to exist; the local conservation law also plays a crucial role in understanding symmetries at the level of local fields and the associated Ward identities.
The argument is rather standard at the classical level (see e.g. Strocchi, F., Nuovo Cimento 42, 9 (1966)) and it can be extended to the quantum case by relying on the quantum action principle of Lowenstein and Lam (Ref. 8). (For a more general justification see Note 3). Clearly in the quantization of a classical Lagrangian a certain arbitrariness is involved in specifying the representation of the quantum operators which correspond to the classical fields. For example, in carrying through the renormalization procedure one may completely destroy the group theoretical structures (like Ward identities, etc.) which characterize the classical case and in such a way possibly define a theory which has very little relation with the classical theory. Such relations are instead maintained in the quantum action principle approach of Lowenstein and Lam and it need not be emphasized here that this is the way to define the quantum version of group theoretical structures which exist at the classical level.
Lowenstein, J., Commun. Math. Phys. 24, 1 (1971)
Lam, Y-M. P., Phys. Rev. D6, 2145, 2161 (1972); D7, 2945 (1973).
See, e.g., the excellent review by Swieca, J., ‘Goldstone Theorem and Related Topics.’ In D. Kastler (ed,). Cargése Lectures in Physics, Vol. 4, New York, Gordon and Breach, 1970.
Ferrari, R., Picasso, L.E., and Strocchi, F., Commun. Math. Phys. 35, 25 (1974).
Strocchi, F., Phys. Rev. D17, 2010 (1978).
For a more precise definition of locality and positivity in QFT see Streater, R. F., and Wightman, A. S., PCT, Spin and Statistics and All That, Benjamin, New York, 1964.
For a more detailed discussion we refer to any standard textbook on quantum electrodynamics, for example Schweber, S., Introduction to Relativistic Quantum Field Theory, Harper and Row, New York, 1964; or Mandl, F., Introduction to Quantum Field Theory, Interscience, New York, 1966.
Strocchi, F., and Wightman, A. S., J. Math. Phys. 15, 2198 (1974).
Ferrari, R., Picasso, L. E., and Strocchi, F., Commun. Math. Phys. 35, 25 (1974).
Maison, D., and Zwanziger, D., Nucl. Phys. 91B, 425 (1975).
Ferrari, R., Picasso, L. E., and Strocchi, F., Nuovo Cimento 39A, 1 (1977).
For a more detailed and rigorous discussion we refer the reader to Ref. 13.
The importance of this property of causality or locality has been emphasized by Haag, R., and Kastler, D., J. Math. Phys. 5, 848 (1964).
The proof has purposely been oversimplified in order not to obscure the underlying physical ideas. Actually, since by the results discussed in Section 3, Gauss’ law can be required to hold only when one takes matrix elements between physical states one should carefully check that this is the case in the above derivation. To this purpose one has to remember that, since ‘longitudinal modes’ are not observable, matrix elements of observable operators taken between vectors of ℋ ’ should not change if one makes a gauge transformation Ф → Ф + X,X being a vector of ℋ ’ containing longitudinal modes. Hence >Ψ, AФ = Ψ,A(Ф + X)>, which implies >Ψ, A Ф> = >Ψ, A (Ф + X)> = 0 For a more detailed discussion we refer the reader to Ref. 15.
For a more detailed discussion see Ref. 5, Sect. IV, where also the relevance of this feature for the solution of the infrared problem is emphasized.
Such a theorem requires the existence of massless particles (Goldstone bosons) with the quantum numbers of the broken generators and there does not seem to be any candidate for such particles, the only experimentally observed massless boson being the photon, which has the feature of being associated to the unbroken internal symmetry group generated by the electric charge.
F. Strocchi, Commun. Math. Phys. 56, 57 (1977).
For a more accurate treatment see Ref. 20.
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Strocchi, F. (1981). Gauss’ Law in Local Quantum Field Theory. In: Tirapegui, E. (eds) Field Theory, Quantization and Statistical Physics. Mathematical Physics and Applied Mathematics, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8368-7_12
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